Score-Based Generative Modeling through Anisotropic Stochastic Partial Differential Equations

2026-05-09

AI Summary

This work addresses the limitation of conventional diffusion models, which excessively disrupt image geometric structures during the forward process, thereby constraining generation quality. It introduces anisotropy into score-based generative modeling for the first time, proposing a class of direction-dependent stochastic partial differential equations (SPDEs). These SPDEs employ structured diffusion coefficients and anisotropic smoothing operators to controllably preserve geometric information over long time scales. By integrating deterministic structure preservation with stochastic perturbations, the method enables direction-adaptive modulation of the information degradation process. Experiments demonstrate that the approach significantly outperforms SDE-based baselines and state-of-the-art flow matching methods in both unconditional image generation and stroke-to-image conditional synthesis, achieving consistent improvements in reconstruction fidelity and image quality metrics across both pixel and latent spaces.

Abstract

Score-based generative modeling (SBGM) has achieved state-of-the-art performance in image generation, with the quality of generated images being highly dependent on the design of the forward (diffusion) process. Among these, models based on stochastic differential equations (SDEs) have proven particularly effective. While traditional methods aim to progressively destroy all image information to enable reconstruction from pure noise, we propose a class of anisotropic stochastic partial differential equations (SPDEs) that preserve the geometric structure of the data over longer time scales throughout the transformation. These SPDEs consist of a drift term that enforces deterministic destruction via structured smoothing, and a diffusion coefficient that enables random destruction through noise injection. Both components are governed by anisotropy coefficients, enabling controlled, direction-dependent information degradation. This framework provides the theoretical foundation for a novel anisotropic score-based generative model. By retaining geometric structure for longer time scales, the backward generative process can exploit residual geometric cues, leading to improved reconstruction fidelity. We empirically validate this improvement in a proof-of-concept implementation on unconditional image generation, showing that anisotropic diffusion can achieve superior image quality metrics. We demonstrate consistent improvements in both pixel and latent space experiments over the SDE-driven baseline as well as over the state-of-the-art Flow Matching approach. Finally, we demonstrate the effectiveness of the introduced anisotropy in a conditional stroke-to-image generation task.

Problem

• score-based generative modeling
• stochastic differential equations
• anisotropic diffusion
• geometric structure preservation
• image generation

Innovation

• anisotropic SPDEs
• score-based generative modeling
• geometric structure preservation
• structured diffusion
• direction-dependent degradation